Topics

• The aim for the model is to predict the House Price in New York

• The data set was original sourced from the mosaic Data package in R

• The data set contains information about New York Houses including prices of Houses (in Us Dollars), lot size of the house (acres), number of bedrooms and bathrooms and the type of heating system

Methodology

• to achieve our aim we perform several models
• then our models were evaluated using RMSE, MAE and adjusted \(r^2\) to find the most useful and appropriate model to use

Data Structure

• it is a large data set with 1,734 rows and 16 columns
• 16 variables, 10 numeric variables and 6 factor variables
• the test variable was removed from the data set because we do not know the meaning of the variable

Rows: 1,734
Columns: 16
$ Price         <int> 132500, 181115, 109000, 155000, 86060, 120000, 153000, 1…
$ Lot.Size      <dbl> 0.09, 0.92, 0.19, 0.41, 0.11, 0.68, 0.40, 1.21, 0.83, 1.…
$ Waterfront    <fct> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ Age           <int> 42, 0, 133, 13, 0, 31, 33, 23, 36, 4, 123, 1, 13, 153, 9…
$ Land.Value    <int> 50000, 22300, 7300, 18700, 15000, 14000, 23300, 14600, 2…
$ New.Construct <fct> 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ Central.Air   <fct> 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,…
$ Fuel.Type     <fct> Electric, Gas, Gas, Gas, Gas, Gas, Oil, Oil, Electric, G…
$ Heat.Type     <fct> Electric, Hot Water, Hot Water, Hot Air, Hot Air, Hot Ai…
$ Sewer.Type    <fct> Private, Private, Public, Private, Public, Private, Priv…
$ Living.Area   <int> 906, 1953, 1944, 1944, 840, 1152, 2752, 1662, 1632, 1416…
$ Pct.College   <int> 35, 51, 51, 51, 51, 22, 51, 35, 51, 44, 51, 51, 41, 57, …
$ Bedrooms      <int> 2, 3, 4, 3, 2, 4, 4, 4, 3, 3, 7, 3, 2, 3, 3, 3, 3, 4, 2,…
$ Fireplaces    <int> 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,…
$ Bathrooms     <dbl> 1.0, 2.5, 1.0, 1.5, 1.0, 1.0, 1.5, 1.5, 1.5, 1.5, 1.0, 2…
$ Rooms         <int> 5, 6, 8, 5, 3, 8, 8, 9, 8, 6, 12, 6, 4, 5, 8, 4, 7, 12, …

Mean, SD, Max, Min, etc.

This data set does not contain any missing data however, outliers do exist in this data set. For example a 0- acre lot size cannot exist. Also, it is unlikely that $5000 USD would be enough to purchase a House in New York in 2006.

General view

Variable chosen

Since our dependent variable is ‘Price’, based on the graph, it can easily found that the variable ‘Living.Area’ may be the most affected by ‘Price’ (0.71, with dark pink).

Dependent variable: Price

Independent variable: Living.Area

Call:
lm(formula = Price ~ Living.Area, data = house)

Coefficients:
(Intercept)  Living.Area  
    12844.2        113.4  

Assumption checking

The residuals \(\varepsilon_i\)are iid\({\mathcal N}(0,\sigma^2)\) and there is a linear relationship between y and x.

Assumption checking

• Linearity: The Auxiliary line is reasonably well plotted like a straight line (with no obvious curve), so there is no obvious pattern in the residual vs fitted values plot.

• Homoskedasticity: It appears the residuals are getting spread-out and do not appear to be fanning out or changing their variability over the range of the fitted values so the constant error variance assumption is met.

• Normality: in the QQ plot, the points are reasonably close to the diagonal line. Although there seems to be some outliers exist, the data-set is relatively large(with 1733 observations). Thus, the normality assumption is approximately satisfied.

Simple regression model

Conclusion：

As the assumption all met, it can be concluded that our simple estimated model is \(\widehat{Price} = 12844.179 + 113.373 \times Living.Area\)

Backward AIC

Assumption Checking for Backward AIC (1)

Assumption Checking for Backward AIC (2)

Forward AIC

Assumption Checking for Forward AIC (1)

Assumption Checking for Forward AIC (2)

Exhaustive Search

Fitted Model

Price ~ Lot.Size + Waterfront1 + Age + Land.Value + New.Construct1 + Central.Air1 + Fuel.TypeGas + Fuel.TypeOil + Heat.TypeHot Water + Heat.TypeNone + Living.Area + Bedrooms + Bathrooms + Rooms

Assumption Checking for the stable model

Our models - simple model

\[\widehat{Price} = 12844.179 + 113.373 \times Living.Area\]

Our models - backward and forward models

Our models - stable model

• Stable: \(\widehat{Price}\) = Waterfront + Land.Value + Living.Area + Bathrooms

r-square and adjusted r-square

• What percentage of the total variation of observed house prices is explained by our models
  • \(r^2\)
  • Adjusted \(r^2\)
    • Take the number of predictors in models into account

• Compare \(r^2\) and adjusted-\(r^2\) of our models:

Error rate

• How different it is between predicted house prices by our models and the actual house prices
  • MAE: mean absolute error
  • RMSE: root mean square error

k-fold cross validation

• \(k\)-fold cross validation estimation¹:

10-fold cross validation

• Data collected by Candice Corvetti and used in the “Stat 101” case study “How much is a Fireplace Worth”. See also https://www.saratogacountyny.gov/departments/real-property-tax-service-agency/
• El Hattab, Hakim, and JJ Allaire. 2017. Revealjs: R Markdown Format for Reveal.js Presentations. https://github.com/rstudio/revealjs.